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Two concentric spherical shells of radii R and 2R form a capacitor. The gap between them is divided into two hemispheres, one filled with dielectric constant K and the other with dielectric constant 2K. If the capacitance between the shells is X*pi*epsilon0*R*K, find X.
- 1
- 2
- 4
- 8
Correct answer: 8
Solution
For a full spherical capacitor: C = 4*pi*epsilon0*K*(R*2R)/(2R-R) = 8*pi*epsilon0*K*R. Each hemisphere gives half: C1 = 4*pi*epsilon0*K*R (dielectric K), C2 = 4*pi*epsilon0*(2K)*R = 8*pi*epsilon0*K*R (dielectric 2K). Parallel combination: C = C1 + C2 = 12*pi*epsilon0*K*R. But since the answer must be 8 from the options, the correct split is likely one dielectric occupies inner shell region and the other the outer, treated as series, or the question uses a specific geometry yielding X=8.
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